Skip to main content
SpringerLink
Log in
Book cover

The Theory of Algebraic Number Fields

  • Book
  • © 1998

Overview

Authors:
  1. David Hilbert
    View author publications

    You can also search for this author in PubMed   Google Scholar

  • This book was tranlated into French many years ago but never into English

  • Hilbert's lucid writing style, bringing out the essentials of the subject is of particular interest now, at a time, when this subject has grown through several decades of intense research activity to one requiring a high degree of technical sophistication

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (36 chapters)

  1. Front Matter

    Pages I-XXXVI
    Download chapter PDF

  2. The Theory of General Number Fields

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Algebraic Numbers and Number Fields

      • David Hilbert
      Pages 3-7

    3. Ideals of Number Fields

      • David Hilbert
      Pages 9-16

    4. Congruences with Respect to Ideals

      • David Hilbert
      Pages 17-23

    5. The Discriminant of a Field and its Divisors

      • David Hilbert
      Pages 25-32

    6. Extension Fields

      • David Hilbert
      Pages 33-39

    7. Units of a Field

      • David Hilbert
      Pages 41-52

    8. Ideal Classes of a Field

      • David Hilbert
      Pages 53-64

    9. Reducible Forms of a Field

      • David Hilbert
      Pages 65-66

    10. Orders in a Field

      • David Hilbert
      Pages 67-75

  4. Quadratic Number Fields

    1. Front Matter

      Pages 113-113
      Download chapter PDF

    2. Factorisation of Numbers in Quadratic Fields

      • David Hilbert
      Pages 115-119
Back to top

Keywords

About this book

Constance Reid, in Chapter VII of her book Hilbert, tells the story of the writing of the Zahlbericht, as his report entitled Die Theorie der algebra is­ chen Zahlkorper has always been known. At its annual meeting in 1893 the Deutsche Mathematiker-Vereinigung (the German Mathematical Society) invited Hilbert and Minkowski to prepare a report on the current state of affairs in the theory of numbers, to be completed in two years. The two mathematicians agreed that Minkowski should write about rational number theory and Hilbert about algebraic number theory. Although Hilbert had almost completed his share of the report by the beginning of 1896 Minkowski had made much less progress and it was agreed that he should withdraw from his part of the project. Shortly afterwards Hilbert finished writing his report on algebraic number fields and the manuscript, carefully copied by his wife, was sent to the printers. The proofs were read by Minkowski, aided in part by Hurwitz, slowly and carefully, with close attention to the mathematical exposition as well as to the type-setting; at Minkowski's insistence Hilbert included a note of thanks to his wife. As Constance Reid writes, "The report on algebraic number fields exceeded in every way the expectation of the members of the Mathemati­ cal Society. They had asked for a summary of the current state of affairs in the theory. They received a masterpiece, which simply and clearly fitted all the difficult developments of recent times into an elegantly integrated theory.

Bibliographic Information

  • Book Title: The Theory of Algebraic Number Fields

  • Authors: David Hilbert

  • DOI: https://doi.org/10.1007/978-3-662-03545-0

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-3-540-62779-1Published: 20 August 1998

  • Softcover ISBN: 978-3-642-08306-8Published: 09 December 2010

  • eBook ISBN: 978-3-662-03545-0Published: 14 March 2013

  • Edition Number: 1

  • Number of Pages: XXXVI, 351

  • Topics: Number Theory, History of Mathematical Sciences

Publish with us

Policies and ethics

Back to top

Search

Navigation